Network Event Data over Time: Prediction and Latent Variable Modeling

Transcription

1 Network Event Data over Time: Prediction and Latent Variable Modeling Padhraic Smyth University of California, Irvine Machine Learning with Graphs Workshop, July 25 th 2010

2 Acknowledgements PhD students: Arthur Asuncion, Chris DuBois, Jimmy Foulds Funding National Science Foundation Office of Naval Research (MURI grant) NDSEG Graduate Fellowship Yahoo!, Google, IBM, Microsoft, Experian Padhraic Smyth: MLG Workshop, KDD 2010: 2

3 Resources A survey of statistical network models A. Goldenberg, A. Zheng, S. Fienberg, E. Airoldi, Foundations and Trends in Machine Learning, 2009 Multiplicative latent factor models for description and prediction of social networks P. D. Hoff, Computational and Mathematical Organization Theory, Random effects models for network data P. D. Hoff, in Dynamic Social Network Modeling and Analysis, 2003 A relational event model for social action C. E. Butts, Sociological Methodology, 2008 Slides from 2010 Whistler Summer School on Social Networks Padhraic Smyth: MLG Workshop, KDD 2010: 3

4 Static Network Data General Notation: N actors (node set) Will assume that set of actors is known and fixed Edges between actors (Y) Adjacency matrix Y y i,j indicates an edge between actor i and actor j Simplest case: binary undirected/directed edges Covariates/Attributes (X) e.g., for each actor (e.g., age, text documents,..) e.g., for each edge (e..g., numeric weights, vector of attributes, text, etc ) Padhraic Smyth: MLG Workshop, KDD 2010: 4

5 Dynamic Network Data Case 1: discrete-time Y t represents the state of the network at discrete time t Data D = {Y 1 Y t. Y T } Example actors = students in a school Y t = friendships between students in month t, t = 1, 12 Interest is often in network dynamics and evolution e.g., Markov models for P( Y t+1 Y t ) Padhraic Smyth: MLG Workshop, KDD 2010: 5

6 Carter Butts Padhraic Smyth: MLG Workshop, KDD 2010: 6

7 Dynamic Network Data Case 2: continuous-time network events y t is an edge between some pair i and j at time t Birth-death edges: each y t has a start and end time Flipping edges: edges can switch on or off Instantaneous edges: each y t is (effectively) instantaneous Data D = { y 1 y t. y T } - in a sense there is no graph Example actors = students in a school y t = between 2 students at time t (would need to allow for multiple recipients ) Interest is often in rates and patterns of communication e.g., Poisson rates for y i,j given network history up to time t Padhraic Smyth: MLG Workshop, KDD 2010: 7

8 Enron Data (Figure from Goldenberg et al, 2010) Padhraic Smyth: MLG Workshop, KDD 2010: 8

9 Time 1 Padhraic Smyth: MLG Workshop, KDD 2010: 9

10 Time 1 Padhraic Smyth: MLG Workshop, KDD 2010: 10

11 Time 2 Padhraic Smyth: MLG Workshop, KDD 2010: 11

12 Time 50 Padhraic Smyth: MLG Workshop, KDD 2010: 12

13 Relational Event Model Butts, 2009 λ (i,j) = Poisson rate of edge generation between actor i and actor j λ t (i,j) = function of network features up to time t, Results in a piecewise constant inhomogeneous Poisson process - rates(t) are a function of network history at time t - between events the rates are constant Typical features include: - individual actor effects - persistence between pairs - preferential attachment - conversational behavior Padhraic Smyth: MLG Workshop, KDD 2010: 13

14 Relational Event Model Example log λ t (i,j) = log λ 0 + log λ i + log λ j + β t x t (i,j) p-dim vector of weights p-dim vector of features Estimation Can be fit with standard regression methods (survival analysis) Likelihood involves O(N 2 ) terms for each of T events Does not scale well Nonetheless an interesting model. See Butts (2008) for an application to emergency response communications Padhraic Smyth: MLG Workshop, KDD 2010: 14

15 Ordinal Version of Relational Events If we don t have time-stamps, but do have the order of the events Can use the fact that choice probability can be written as P(i, j) = λ t (i,j) / Σ λ t (i,j) Can still learn the model from sequence of events, with relative rates Overall network rate λ 0 is unspecified Padhraic Smyth: MLG Workshop, KDD 2010: 15

16 Additional Modeling Aspects. As with static networks. Actor attributes, e.g., actor age Edge (event) attributes, e.g., text of an Can also have time-dependent covariates/attributes E.g., actor attributes changing over time Network level external covariates Calendar effects: time of day, day of week, time of year External events exogenous time-series Padhraic Smyth: MLG Workshop, KDD 2010: 16

17 Outline of the Talk Begin with statistical models for static data In particular, latent variable models Review some useful approaches in this area Look at how to extend these models to temporal data Particularly relational event data Discuss recent work [Caveat: only focus on certain approaches, not exhaustive] Evaluation and prediction Some general comments Mostly review.with some new work towards the end Padhraic Smyth: MLG Workshop, KDD 2010: 17

18 Why Statistical Modeling? Learning can estimate network properties from data in a principled way Prediction/Querying reduces to computation of relevant conditional probabilities and expectations Noise/Missing Data Systematic way to handle real-world noise Covariates Relatively straightforward to integrate non-network information into the network model Padhraic Smyth: MLG Workshop, KDD 2010: 18

19 Slide from Dave Hunter Padhraic Smyth: MLG Workshop, KDD 2010: 19

20 Estimation is Hard P(G θ) = f( G ; θ ) / normalization constant The normalization constant = sum over all possible graphs Say binary directed graphs: how many graphs? 2 n(n-1) e.g., with n = 50, we will have ~ graphs to sum over MCMC techniques are now the method of choice but many problems with degeneracy of likelihoods difficult models to fit (e.g., see Robins et al, Social Networks, 2007) Padhraic Smyth: MLG Workshop, KDD 2010: 20

21 An Alternative: Latent Variable Models ERGMs allow us to model edge dependencies in very flexible ways but the computational penalty is too high Latent Variable Models Typically, latent variables are chosen so that edges are conditionally independent given the latent variables Can lead to much simpler models than full ERGMs If we can find useful and tractable latent variable representations, this may provide a good alternative to ERGMs Padhraic Smyth: MLG Workshop, KDD 2010: 21

22 Idea: Example: The Latent Space Model Embed nodes in a latent K-dimensional Euclidean space Probability of edge (i,j) = f (distance (i, j) ) Edges are conditionally independent given K-dim locations Hoff, Raftery, Handcock, 2002 Padhraic Smyth: MLG Workshop, KDD 2010: 22

23 Idea: Example: The Latent Space Model Embed nodes in a latent K-dimensional Euclidean space Probability of edge (i,j) = f (distance (i, j) ) Edges are conditionally independent given K-dim locations Probability model z i = K-dim latent position vector for node i Log-odds (y ij = 1) = log P( y ij = 1)/(1 P( y ij = 1) ) Hoff, Raftery, Handcock, 2002 = - z i - z j + µ + β x ij covariate effects (optional) distance of nodes i and j network density parameter Padhraic Smyth: MLG Workshop, KDD 2010: 23

24 Likelihood: Estimation: Example: The Latent Space Model P(Y Z, b, m) = Π P( y ij z i, z j, µ, β ) Can maximize likelihood directly (as a function of Z,.) using gradient methods Can also be Bayesian, use priors, and sample from posterior density using MCMC Can also introduce block/cluster structure on nodes (see Handcock, Raftery, and Tantrum, 2007) Computational issues logistic function Hoff, Raftery, Handcock, 2002 Note that the product above is over all pairs, O(N 2 ): poor scalability Recent work (Raftery et al, 2010) shows how to ignore many non-edges Padhraic Smyth: MLG Workshop, KDD 2010: 24

25 Figure from Hoff, Raftery, Handcock, 2002 Padhraic Smyth: MLG Workshop, KDD 2010: 25

26 Figure from Hoff, Raftery, Handcock, 2002 Representational issue Is Euclidean space embedding a good way to represent network information? Similar to issues with multidimensional scaling Padhraic Smyth: MLG Workshop, KDD 2010: 26

27 Example: Relational Topic Model Chang and Blei, 2009 Nodes = documents, edges = links between documents Standard LDA/topic model, but.. Topics i,j influence p(edge i, j) Edge(i, j) influences topics i and j Model is similar to latent-space model Latent space: actors represented by k-dim location Relational topics: docs represented by k-dim topic distribution Both use logistic-like links for edge probabilities Padhraic Smyth: MLG Workshop, KDD 2010: 27

28 Relational Topic Model (RTM) [Chang, Blei, 2009] Same setup as LDA, except we have observed network information across documents (adjacency matrix) α Link probability function θ kd η,υ θ kd' Z id y d, d' β Z id' Documents with similar topics are more likely to be linked Topics influence links, and links infuence topics X id N d Φ wk K X id' N d Padhraic Smyth: MLG Workshop, KDD 2010: 28

29 Exponential: Link probability functions Logistic: K-dim vector of topic proportions Normal CDF: Normal: where Element-wise product Padhraic Smyth: MLG Workshop, KDD 2010: 29

30 Link Prediction with Wikipedia Movie Pages 'Sholay' Indian film, 45% of words belong to topic 24 (Hindi topic) Top 5 most probable movie links in training set: 'Laawaris 'Hote Hote Pyaar Ho Gaya 'Trishul 'Mr. Natwarlal 'Rangeela Cowboy Western film, 25% of words belong to topic 7 (western topic) Top 5 most probable movie links in training set: 'Tall in the Saddle 'The Indian Fighter' 'Dakota' 'The Train Robbers' 'A Lady Takes a Chance Rocky II Boxing film, 40% of words belong to topic 47 (sports topic) Top 5 most probable movie links in training set: 'Bull Durham '2003 World Series 'Bowfinger 'Rocky V 'Rocky IV' Padhraic Smyth: MLG Workshop, KDD 2010: 30

31 Idea: Example: Stochastic Block Model Partition the set of nodes into K blocks that are structurally equivalent Model interactions at the K x K block level instead of N x N actor level P( y ij ) = P( y ki, kj ), k i, k j ε {1, K} e.g., Nowicki and Snijders, 2001 Padhraic Smyth: MLG Workshop, KDD 2010: 31

32 Idea: Example: Stochastic Block Model Partition the set of nodes into K blocks that are structurally equivalent Model interactions at the K x K block level instead of N x N actor level P( y ij ) = P( y ki, kj ), k i, k j ε {1, K} e.g., Nowicki and Snijders, 2001 (Figure from Goldenberg et al, 2010) Padhraic Smyth: MLG Workshop, KDD 2010: 32

33 Example: Stochastic Block Model e.g., Nowicki and Snijders, 2001 Estimation 2 sets of parameters 1. B = block-level interaction matrix, e.g., K x K matrix of Bernoullis 2. Z = N indicator variables, mapping each node to one of K blocks (Can use your favorite estimation technique: EM, gradient, MCMC, etc) - See also Infinite Relational Model (IRM), Kemp et al (2006) - Allows one to learn the number of blocks Padhraic Smyth: MLG Workshop, KDD 2010: 33

34 Mixed Membership Stochastic Block Model (MMB) Airoldi, et al, 2008 Generalizes the stochastic block model to allow mixed membership Specifically, Replace N indicator variables, with N multinomials z i, i = 1, N Each multinomial is a distribution over the K blocks Allows an actor to have multiple memberships, with prob z i1, z ik Padhraic Smyth: MLG Workshop, KDD 2010: 34

35 Mixed Membership Stochastic Block Model (MMB) Airoldi, et al, 2008 Generalizes the stochastic block model to allow mixed membership Specifically, Replace N indicator variables, with N multinomials z i, i = 1, N Each multinomial is a distribution over the K blocks Allows an actor to have multiple memberships, with prob z i1, z ik Generative model For each actor: multinomial z i ~ Dirichlet For each possible edge: Estimation k i ~ multinomial z i, k j ~ multinomial z j y ij, ~ B( k i, k j ) Likelihood involves O(N 2 ) terms: can use variational or MCMC methods Padhraic Smyth: MLG Workshop, KDD 2010: 35

36 Mixed Membership Stochastic Blockmodel Stochastic Blockmodel Figures from Airoldi, et al, 2008 Padhraic Smyth: MLG Workshop, KDD 2010: 36

37 Binary Feature Relational Model Miller, Griffiths, Jordan, 2009 Based on idea of Indian Buffet Process (Griffiths and Ghahramani, 2006) Represent each object by a set of latent binary features Learn binary features that explain well the observed data Non-parametric: infinite number of features.but in practice, given data, only a finite number are inferred Motivation: Classes defined over combinatorial number of binary features Different from MMB, e.g., male high school musicians/athletes Different from latent space Can apply this idea to network data Latent variable model where p(edge i, j) is a function of i and j s latent binary features Padhraic Smyth: MLG Workshop, KDD 2010: 37

38 Relational Binary Feature Model for Networks Miller, Griffiths, Jordan, 2009 Hidden Features Actors Presence of edge between actor i and actor j is (e.g.) a logistic function of a weighted sum of features they have in common Estimation: based on MCMC Figure from Griffiths and Ghahramani, 2006 Padhraic Smyth: MLG Workshop, KDD 2010: 38

39 Predictions on NIPS Coauthorship Data From Miller, Griffiths, Jordan, 2009 Padhraic Smyth: MLG Workshop, KDD 2010: 39

40 A Unified View P ( y ij = 1) = f ( g( z i, z j ) + β x ij + µ ) (see also Hoff, 2009; Airoldi 2010) where f = logistic function (for example) z i, z j = k x 1 latent vectors for the ith and jth nodes g = function that combines latent vectors, with parameters θ x i j = covariate vector for the pair of nodes µ = network density parameter Padhraic Smyth: MLG Workshop, KDD 2010: 40

41 Examples Log-odds ( y ij = 1) = g( z i, z j ) + β x ij + µ Latent space model: z i, z j = k x 1 vectors of latent positions in Euclidean space g( z i, z j ) = - z i - z j Padhraic Smyth: MLG Workshop, KDD 2010: 41

42 Examples Log-odds ( y ij = 1) = g( z i, z j ) + β x ij + µ Latent space model: z i, z j = k x 1 vectors of latent positions in Euclidean space g( z i, z j ) = - z i - z j Latent factor model: (see Hoff, 2008) z i = k x 1 real-valued vector g( z i, z j ) = z i W z j, where W is a k x k diagonal matrix Padhraic Smyth: MLG Workshop, KDD 2010: 42

43 Examples Log-odds ( y ij = 1) = g( z i, z j ) + β x ij + µ Latent space model: z i, z j = k x 1 vectors of latent positions in Euclidean space g( z i, z j ) = - z i - z j Latent factor model: (see Hoff, 2008) z i = k x 1 real-valued vector g( z i, z j ) = z i W z j, where W is a k x k diagonal matrix Relational topic model: z i = k-dimensional topic distribution (multinomial) for document i g( z i, z j ) = weighted element-wise product of the 2 topics Padhraic Smyth: MLG Workshop, KDD 2010: 43

44 Examples Log-odds ( y ij = 1) = g( z i, z j ) + β x ij + µ Latent class or stochastic blockmodel: z i = fixed k-dimensional binary indicator vector, e.g., (0, 0, 1, 0, 0) g( z i, z j ) = W zi, zj, where W is a k x k matrix The indicators select which element (block) to use Padhraic Smyth: MLG Workshop, KDD 2010: 44

45 Examples Log-odds ( y ij = 1) = g( z i, z j ) + β x ij + µ Latent class or stochastic blockmodel: z i = fixed k-dimensional binary indicator vector, e.g., (0, 0, 1, 0, 0) g( z i, z j ) = W zi, zj, where W is a k x k matrix The indicators select which element (block) to use Mixed membership stochastic blockmodel (MMB) Like latent class, but z i = sampled from actor multinomial i Padhraic Smyth: MLG Workshop, KDD 2010: 45

46 Examples Log-odds ( y ij = 1) = g( z i, z j ) + β x ij + µ Latent class or stochastic blockmodel: z i = fixed k-dimensional binary indicator vector, e.g., (0, 0, 1, 0, 0) g( z i, z j ) = W zi, zj, where W is a k x k matrix The indicators select which element (block) to use Mixed membership stochastic blockmodel (MMB) Like latent class, but z i = sampled from actor multinomial i Relational binary feature model (finite version): z i = k-dimensional binary vector, e.g., (1, 0, 1, 0, 1) g( z i, z j ) = z i W z j, where W is a k x k matrix The combination of on features determine the pairwise effect Padhraic Smyth: MLG Workshop, KDD 2010: 46

47 Adding Time. General static form for latent variable models: Log-odds ( y ij = 1) = g( z i, z j ) + β x ij + µ One approach is to make the z s time-dependent i.e., allow latent features of each actor change over time An example: Gaussian linear motion models in z-space - Sarkar and Moore (2005) for actors latent-space positions - Fu, Song, and Xing (2009) for actors mixed membership vectors Padhraic Smyth: MLG Workshop, KDD 2010: 47

48 Event Data and Latent Variables Data = series of time-stamped binary directed events among actors 2 processes we want to model: Rates (e.g., Poisson) Choices (who connects to who) A simple approach: Each pair i, j (at time t) has an event rate that is Poisson λ ij Global network rate = Σ λ ij = λ P( y ij ) = λ ij / λ or, λ ij = P( y ij ) x λ Here P( y ij ) is a multinomial with O(N 2 ) entries : given that an event will happen, which pair will it be? (different from binary y ij variables we saw before) Padhraic Smyth: MLG Workshop, KDD 2010: 48

49 Direct Estimation We could predict the likelihood of i and j communicating based directly on i and j s history Multinomial with O(N 2 ) entries Can use smoothing to combat sparsity Problems Data can be extremely sparse for large N smoothing is non-informative, and does not borrow strength from the graph Nonetheless this is a useful baseline when evaluating predictions Historically, few papers evaluate models predictively Even fewer compare their models to simple baselines Padhraic Smyth: MLG Workshop, KDD 2010: 49

50 Illustration of Sparsity: Frequency of Events per pair of Actors International Political Events data King, 2003 Padhraic Smyth: MLG Workshop, KDD 2010: 50

51 Mixtures for Relational Events Talk by Chris DuBois, Tuesday Mixture model over events First choose event class k, k = 1,. K k ~ π y ij : i ~ φ ( sender nodes k), j ~ φ ( receiver nodes k) Parameters π : K x 1 multinomial = relative likelihood of different event classes φ ( sender nodes k), φ ( receiver nodes k) 2K multinomials, each of size N Simple model Similar to model proposed by Sinkkonen et al, MLG 2008 Quite similar in spirit to LDA/topic model for documents Padhraic Smyth: MLG Workshop, KDD 2010: 51

52 Marginal Product Mixture Model (MPMM) Likelihood Estimation Can use EM or Collapsed Gibbs Sampling Both are fast only need to loop over observed events (can ignore pairs where no events occurred) Extensions Modulate choice process with time-varying network rate Different types of events ( actions ) Markov (hidden) dependence on selection of event class Padhraic Smyth: MLG Workshop, KDD 2010: 52

53 MMB model Comparing MPMM and MMB For every pair of actors Sample latent class for i and j Given latent classes, sample a binary edge, or a count (e.g., Poisson) MPMM model For every event Select latent class of event Given latent class, sample i and j Differences MMB models whole graphs, but not individual events So dynamics are from graph to graph (e.g., Fu, Song, Xing 2009) MPMM models individual events, not whole graphs Allows dynamics at the event level (e.g., Markov dependence of events) And inference in MPMM is much more tractable. Padhraic Smyth: MLG Workshop, KDD 2010: 53

54 Estimation EM Straightforward and fast MCMC, Collapsed Gibbs sampling Also straightforward and fast Both EM and Gibbs scale linearly in the number of observed events (edges) Easy to apply to large data sets Padhraic Smyth: MLG Workshop, KDD 2010: 54

55 Eckmann Data Set 200,000 s 2997 individuals, 82 days Padhraic Smyth: MLG Workshop, KDD 2010: 55

56 Data (Eckmann) Padhraic Smyth: MLG Workshop, KDD 2010: 56

57 International Relations Data (King, 2003) 40,000 events 2700 actors 171 action types Padhraic Smyth: MLG Workshop, KDD 2010: 57

58 Padhraic Smyth: MLG Workshop, KDD 2010: 58

59 Prediction and Evaluation Use future data to evaluate predictive power and compare models Metrics Log score = log probability of events that actually occurred Brier/MSE style scores Ranking/ROC scores Padhraic Smyth: MLG Workshop, KDD 2010: 59

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62 Comments on Evaluation Prediction on independent test data is critical Relatively easy to do with dynamic networks Tricky to do with static networks (but see Hoff, 2009) Caveat For link (or link probability) prediction it can be very difficult to beat relatively simple baselines, e.g., Graph(t+1) = Graph(t) p(event) = smoothed estimate based on historical frequency of that pair Solution? More interesting questions than just predicting what happens next, e.g How likely is that group A will communicate with group B in the next k days? If we have events with missing information, can we infer sender/receiver? Can we detect significant shifts/non-stationarity? Padhraic Smyth: MLG Workshop, KDD 2010: 62

63 What Next? Historically, social science applications of network analysis focused on understanding rather than prediction per se For data miners/computer scientists, predictive modeling plays a much more important role Key question: what are the important applications/problems that network/graph models can solve, that can t be solved by other means? Candidates? e.g., tools for egocentric modeling/analysis/management of personal communication data ( , social media, etc) Change detection Ranking of incoming communication events Padhraic Smyth: MLG Workshop, KDD 2010: 63

64 Summary Latent variable models are useful for network modeling/prediction Broad toolbox of building blocks May be scalable to large data sets Latent models for dynamic data show promise Dynamic network data comes in multiple forms Aggregated/longitudinal data Time-stamped event data (quite different in nature) Models need to be evaluated via prediction on test data Padhraic Smyth: MLG Workshop, KDD 2010: 64

65 Resources A survey of statistical network models A. Goldenberg, A. Zheng, S. Fienberg, E. Airoldi, Foundations and Trends in Machine Learning, 2009 Multiplicative latent factor models for description and prediction of social networks P. D. Hoff, Computational and Mathematical Organization Theory, Random effects models for network data P. D. Hoff, in Dynamic Social Network Modeling and Analysis, 2003 A relational event model for social action C. E. Butts, Sociological Methodology, 2008 Slides from 2010 Whistler Summer School on Social Networks Padhraic Smyth: MLG Workshop, KDD 2010: 65